Answered Essay: Consider the mechanical system shown in Figure P2.15, p. 43 of Close, Frederick and Newell, Modeling and Analysis of Dynamic

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2. [13 marks Consider the mechanical system shown in Figure P2.15, p. 43 of Close, Frederick and Newell, Modeling and Analysis of Dynamic Systems, 3rd ed., where the springs are neither stretched nor compressed when x1 = x2 = 0. This system is modelled by the equations (a) Define appropriate state variables and write down the matrix form of the state variable equations with inputs fa(t) and g. Also write down the matrix form of the output equations, where the outputs are the force fK = K1(x1-x2) and the acceleration aM2 of M2. (b) Let M1 = M2 = K1 = K2 = B = 1 in this problem. Derive the input-output equation relating xi to the inputs fa(t) and g.

Consider the mechanical system shown in Figure P2.15, p. 43 of Close, Frederick and Newell, Modeling and Analysis of Dynamic Systems, 3rd ed., where the springs are neither stretched nor compressed when x_1 = x_2 = 0. This system is modelled by the equations M_1 x_1 + B x_1 + K_1 x_1 – B x_2 – K_1 x_2 = M_1 g – B x_1 – K_1 x_1 + M_2 x_2 + B x_2 + (K_1 + K_2)x_2 = M_2 g + f_a(t) (a) Define appropriate state variables and write down the matrix form of the state variable equations with inputs f_a(t) and g. Also write down the matrix form of the output equations, where the outputs are the force f_K_1 = K_1(x_1 – x_2) and the acceleration a_M_2 of M_2. (b) Let M_1 = M_2 = K_1 = K_2 = B = 1 in this problem. Derive the input-output equation relating xi to the inputs f_a(t) and g.

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